TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 9, SỐ 9 -2006 Trang 27 EXISTENCE OF SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS WITH SINGULAR CONDITIONS Chung Nhan Phu, Tran Tan Quoc University of Natural Sciences, VNU-HCM (Manuscript Received on March 24 th , 2006, Manuscript Revised October 2 nd , 2006) ABSTRACT: In this paper, we study the existence of generalized solution for a class of singular elliptic equation: ( ) ( ) ( ) ( ) ( ) ( ) diva x,u x , u x f x,u x , u x 0 − ∇+ ∇=. Using the Galerkin approximation in [2, 10] and test functions introduced by Drabek, Kufner, Nicolosi in [5], we extend some results about elliptic equations in [2, 3, 4, 6, 10]. 1.INTRODUCTION The aim of this paper is to prove the existence of generalized solutions in ( ) 1, 0 p W Ω for the quasilinear elliptic equations: ( ) ( ) ( ) ( ) ( ) ( ) diva x,u x , u x f x,u x , u x 0−∇+∇= (1.1) i.e. proving the existence of ( ) 1,p 0 uW ∈ Ω such that ( ) ( ) ( ) ( ) ( ) ( ) ( ) c a x,u x , u x dx f x,u x , u x dx 0, C ∞ ΩΩ ∇ ∇ϕ + ∇ ϕ = ∀ϕ∈ Ω ∫∫ where Ω is a bounded domain in N ,N 2≥ with smooth boundary, ( ) p1,N∈ and NN N a: ,f:Ω× × → Ω× × → satisfy the following conditions: Each ( ) i ax,,ηξ is a Caratheodory function, that is, measurable in x for any fixed ( ) N1 , + ζ= ηξ ∈ and continuous in ζ for almost all fixed x ∈ Ω , ()() () p1 i1 1 ax,, cx kx,i1,N α− ⎡⎤ ηξ ≤ η +ξ + ∀= ⎣⎦ (1.2) () ( ) ** ax,, ax,, 0 ⎡⎤ ⎡⎤ η ξ− ηξ ξ−ξ > ⎣⎦ ⎣⎦ (1.3) () p ax,, η ξξ≥λξ (1.4) a.e. *N * x, ,, , ∈ Ω∀η∈ ∀ξξ ∈ ξ≠ξ. where ( ) ( ) [ ] p' 1loc 1 1 cL ,c0,kL , 0,p1, 0 ∞ ∈Ω≥∈Ωα∈−λ>. and N f:Ω× × → is a Caratheodory function satisfying ()() () 22 fx,, c x k x βγ ⎡ ⎤ ηξ ≤ η +ξ + ⎣ ⎦ (1.5) () () qr 3 fx,, c x b d η ξη≥− −η−ξ (1.6) where c 2 is a positive function in ( ) loc 3 L,c ∞ Ω is a positive function in ( ) L ∞ Ω , ( ) p' 2 kL∈Ω and r,q [0,p)∈ , b, d are positive constants, [ ] * 0,p 1 , [0,p 1)γ∈ − β∈ − with * Np p Np = − . Science & Technology Development, Vol 9, No.9- 2006 Trang 28 Because ( ) 12 loc c,c L ∞ ∈Ω we cannot define operator on the whole space ( ) 1, 0 p W Ω . Therefore, we cannot use the property of (S + ) operator as usual. To overcome this difficulty, in every n Ω we find solution ( ) 1,p n0n uW ∈ Ω of the equation: ( ) ( ) ( ) ( ) ( ) ( ) diva x,u x , u x f x,u x , u x 0−∇+∇= where { } n Ω is an increasing sequence of open subsets of Ω with smooth boundaries such that n Ω is contained in n1 + Ω and n1 n ∞ = Ω =ΩU . In this case, we only have the strong convergence of { } n u to u in ( ) 1, Ω p loc W by using the same technique of Drabek, Kufner, Nicolosi (in [5], section 2.4). However, it is enough to get the generalized solution. An example for our conditions: () () () () () () () p1 ii11i ab 2 1 ax,, A kxsgn dx 1 fx,, k x sgn dx − θ μ ⎡⎤ η ξ= ξ + η+ ξ ⎣⎦ ⎡⎤ ηξ = ξ +η + η ⎣⎦ where ( ) ( ) 112 dx distx, ;, 0;A,k,k=∂Ωθμ> are positive functions () () [ ] p' * 12 1 k,k L ;A ; ,a 0,p 1;b [0,p 1). α ∈Ω η≤ηα∈ − ∈ − The problem is singular because () () () loc 11 ,L dxdx ∞ θμ ∈ Ω . Remark: 1) If ( ) 2 cL ∞ ∈Ω and ,[0,p1)βγ∈ − the condition (1.5) implies the condition (1.6). 2) The pseudo-Laplacian () ( ) p2 p2 11 N N a x, , , , −− η ξ= ξ ξ ξ ξ , the p-Laplacian () ( ) p2 p2 1N a x, , , , −− ηξ = ξ ξ ξ ξ are some special cases that satisfy our conditions. So our results generalized the corresponding Dirichlet problems in [3, 4]. Our paper also extends the recent result about singular elliptic equations for case p=2 in [6]. 2. PREREQUISITES 2.1.Lemma 2.1 (See e.g. [10], Proposition 1.1, page 3) Let G be a measurable set of positive measure in n and m h:G×× → satisfy the following conditions: a) h is a Caratheodory function. b) () () i m p/p' 1m i i1 h x, u , , u c u g x , x G = ≤+∀∈ ∑ where c is a positive constant, ( ) i p 1, , i 1, ,m∈∞∀= , ( ) p' gLG∈ . Then the Nemytskii operator defined by the equality ( ) ( ) ( ) ( ) ( ) 1m 1 m H u , ,u x h x,u x , ,u x= acts continuously from ( ) ( ) 1m pp L G L G×× to ( ) p' LG. Moreover, it is bounded, i.e. it transforms any set which is TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 9, SỐ 9 -2006 Trang 29 bounded into another bounded set. (Proof of this fact for the simple case can be found in [8], theorem 2.2, page 26) . 2.2.Lemma 2.2 (See e.g. [10], lemma 4.1, page 14) Let m F:U→ be a continuous mapping of the closure of a bounded domain m U ⊂ . Suppose that the origin is an interior point of D and that the condition () () () m ii i1 Fx,x F xx 0, x U = = ≥∀∈∂ ∑ (1.7) Then the equation F(x) =0 has at least one solution in U . We recall some results about Schauder bases. Definition: A sequence { } i x in a Banach space X is a Schauder basis if every xX ∈ can be written uniquely n ii ii n i1 i1 xcxlimcx ∞ →∞ == == ∑ ∑ , where { } i c ⊂ . Because every x X ∈ is written uniquely ii i1 xcx ∞ = = ∑ we have i x0≠ and c i is a function from X to , for all i in R. 2.3.Lemma 2.3: ([9], Theorem 3.1, page 20) For all i in , c i is a continuous linear function on X, i.e. ( ) ii i X i,M0,cxMx,xX∀∈ ∃ > ≤ ∀ ∈ 2.4.Lemma 2.4: ([7], Corollary 3) Let D be a bounded domain in N with smooth boundary. Then the space ( ) 1,p 0 WD has a Schauder basis. 2.5.Lemma 2.5: Let D be an open set in Ω , D ⊂Ω . If weak n uu ⎯ ⎯⎯→ in ( ) 1,p WD (1.8) and ( ) ( ) [ ] nn n n n D lim a x,u , u a x,u , u u u dx 0 →∞ ∇− ∇ ∇−∇ = ⎡⎤ ⎣⎦ ∫ (1.9) Then there exists a subsequence of { } n u still denoted by { } n u such that n uu∇→∇ in ( ) p LD . Proof: Since ( ) 12 loc c,c L ∞ ∈Ω we have ( ) 12 c,c L D ∞ ∈ and the conditions (1.2), (1.5) become: () () p1 i1 1 ax,, C kx,i1,N α− ⎡⎤ ηξ ≤ η +ξ + ∀ = ⎣⎦ () () 22 fx,, C k x βγ ⎡ ⎤ ηξ ≤ η +ξ + ⎣ ⎦ Using the well-known result in [2], Lemma 3, we obtain our Lemma. Let us recall the definition of class (S+): A mapping * T:X X→ is called belongs to the class (S+) if for any sequence u n in X with weak n uu→ and nn n limsup Tu ,u u 0 →∞ −≤ it follows that n uu→ . 2.6.Lemma 2.6: (see [2, 10]) Let D be an open set in Ω , D ⊂Ω and A be a mapping from ( ) 1,p 0 WD to () * 1,p 0 WD ⎡⎤ ⎣⎦ , such that () () N i i1 i DD v Au,v ax,u,u dx fx,u,uvdx x = ∂ =∇+∇ ∂ ∑ ∫∫ Science & Technology Development, Vol 9, No.9- 2006 Trang 30 Then A is a (S + ) operator. 3 . MAIN RESULTS Let { } n Ω be an increasing sequence of open subsets of Ω with smooth boundaries such that n Ω is contained in n1 + Ω and n1 n ∞ = Ω =ΩU . First, in every n Ω we find solution ( ) 1,p n0n uW ∈ Ω of the equation: ( ) ( ) ( ) ( ) ( ) ( ) diva x,u x , u x f x,u x , u x 0−∇+∇= (3.1) Applying the same technique as in [10], Theorem 4.1, page 14, we can show that (3.1) has a bounded solution in ( ) 1,p 0n W Ω . 3.1.Lemma 3.1: For each n Ω , the equation: ( ) ( ) ( ) ( ) ( ) ( ) diva x,u x , u x f x,u x , u x 0 − ∇+ ∇= (3.2) has a solution ( ) 1,p n0n uW∈Ω. Furthermore, there exists a positive constant R independent of n satisfying that () 1,p n 0 n W uR,n Ω ≤ ∀∈ . Proof: Fix n ∈ . Let ( ) 1,p n0 D,XWD=Ω = and A be a mapping from ( ) 1,p 0 WD to () * 1,p 0 WD ⎡⎤ ⎣⎦ , such that () () () N 1,p i 0 i1 i DD v Au,v ax,u,u dx fx,u,uvdx,u,v W D x = ∂ =∇+∇∀∈ ∂ ∑ ∫∫ By Lemma 2.6, A belongs to class (S+). We will prove that A is a demicontinuous operator, i.e. if m uu→ in ( ) 1,p 0 WD , then ( ) 1,p m0 Au , v Au,v , v W D→∀∈ By m uu→ in ( ) 1,p 0 WD and (1.2), (1.5), applying Lemma 2.1, we get ( ) ( ) imm i a.,u,u a.,u,u,i1, ,N∇→ ∇∀= In ( ) p' LD as m →∞ and ( ) ( ) mm f .,u , u f .,u, u∇→ ∇ in ( ) p' LD as m →∞ Hence () () N m i mm mm i1 i DD v Au ,v a x,u , u dx f x,u , u vdx x = ∂ =∇+∇→ ∂ ∑ ∫∫ () () () N 1,p i 0 i1 i DD v a x,u, u dx f x,u, u vdx Au,v , v W D x = ∂ ∇+∇=∀∈ ∂ ∑ ∫∫ Therefore, A is demicontinuous. Besides, by applying the boundedness of Nemytskii operator for ( ) a.,u, u∇ and ( ) f.,u, u ∇ one deduces that A is bounded. For any arbitrary u in ( ) 1,p 0 WD , due to (1.4), (1.6), we have ( ) ( ) DD Au,u a x,u, u udx f x,u, u udx=∇∇+∇ ∫∫ () () () () pqr 3 DD ux dx c x b.ux d. ux dx ⎡ ⎤ ≥λ ∇ − + + ∇ ⎣ ⎦ ∫∫ () () () q r pq 3 XLD LD D uc bu duxdx ∞ ≥λ − − − ∇ ∫ Let ( ) ( ) ux ux, x D=∀∈ ) and ( ) ux 0, x \D=∀∈Ω ) , we have TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 9, SỐ 9 -2006 Trang 31 () () () q r/p 1 r/p q p p 3 XLD L DD Au,u u c b u d u x dx dx ∞ − Ω ⎛⎞⎛⎞ ≥λ − − − ∇ ⎜⎟⎜⎟ ⎝⎠⎝⎠ ∫∫ ) Since * qpp<< , the continuous imbedding ( ) ( ) 1,p q 0 WL Ω →Ω implies that () () ( ) () p 1,p 0 q r p 3 XL LD W Au,u u c b M. u d.K u ∞ Ω Ω ≥λ − − − ∇ ) () () () 1,p 1,p 00 pqr q 3 XL WD WD uc bM.u d.Ku ∞ Ω ≥λ − − − () q 3 p L ppqpr X XXX c b Md.K u uuu ∞ Ω −− ⎛⎞ ⎜⎟ ≥λ− − − ⎜⎟ ⎝⎠ Since 1, r, q<p, one can choose a positive constant R independent of n such that ( ) X Au,u 0, u B 0, R≥∀∈∂ (3.3) Applying Lemma 2.4 there exists a Schauder basis { } i v in the space X. We consider in m the domain () m m1mii i1 X U c c , ,c : c v R = ⎧⎫ == < ⎨⎬ ⎩⎭ ∑ Applying Lemma 2.3, there exists () m iiijj i 1mm j1 x M 0, c M c v M R, i 1,m, c , ,c U = >≤ <∀=∀ ∈ ∑ So U m is bounded in m . We apply Lemma 2.2 to this domain U m and to the mapping m m F:U → , ( ) ( ) ( ) ( ) 1m F c F c , , F c= , () m ijji j1 Fc A cv ,v = ⎛⎞ = ⎜⎟ ⎝⎠ ∑ Let ( ) 1m m c c , ,c U=∈∂ and m jj j1 ucv = = ∑ then X uR = . We have () () () mmm j j jj jj j1 j1 j1 Fc,c F cc A cv , cv Au,u 0 === ⎛⎞ == =≥ ⎜⎟ ⎝⎠ ∑∑∑ because of (3.3). By Lemma 2.2, the equation F(c) =0 has at least one solution in m U , for example c= (c 1 ,…, c m ). Hence () m ijji j1 Fc A cv ,v 0,i 1,m = ⎛⎞ ==∀= ⎜⎟ ⎝⎠ ∑ Consequently, m mjj j1 ucv = = ∑ satisfies the inequality m X uR ≤ (3.4) And is a solution of the system mi Au , v 0, i 1,m=∀= (3.5) Let m go through we have a sequence { } m u satisfying (3.4) and is a solution of (3.5). By virtue of the reflexivity of the space X, the sequence u m contains weakly convergent Science & Technology Development, Vol 9, No.9- 2006 Trang 32 subsequence k m u . So k weak m0 uu→ . Since 0 u is in X with the Schauder basis { } i v , we have m 0jj jj m j1 j1 uvlimv ∞ →∞ == =α= α ∑∑ . Let m mjj j1 wv = = α ∑ then m0 wu→ so k m0 wu→ . We have kk kk k k k mm 0 mm m m m 0 Au , u u Au ,u w Au , w u−= − + − (3.6) Moreover, kk mm 0 k lim Au , w u 0 →∞ − = (3.7) because of (3.4), the boundedness of the operator A, and the strong convergence of k m w to u 0 . Since k kk m mm jj j1 uw v = −=β ∑ and (3.5), we get kk k mm m Au , u w 0, k − =∀. Hence kk mm 0 k lim Au ,u u 0 →∞ − = (3.8) Because A belongs to class (S+) and (3.8), we deduce that k m0 uu→ . Since A is demicontinuous, passing to limit the equality (3.5) for a fixed i, we have 0i Au , v 0 = (3.9) Let vX∈ , then m jj jj m j1 j1 vvlimv ∞ →∞ == =α= α ∑∑ . Since i is an arbitrary index, it follows from (3.9) that m 0ii i1 Au , v 0, m = α=∀∈ ∑ . Let m tend to infinity, we get 0 Au , v 0= Hence, u 0 is a solution of the equation (3.2). Moreover, since k weak m0 uu→ , we get () 1,p k n 0 00 m WX X k uuliminfuR Ω →∞ =≤ ≤ , where R does not depend on n. This completes the proof of Lemma 3.1 By Lemma 3.1, we have proved that (3.2) has a bounded solution ( ) 1,p n0n uW∈Ω satisfying () 1,p n 0 n W uR,n Ω ≤∀∈ . Next, we expand u n to all ( ) nn :u x 0, x \ Ω =∀∈ΩΩ. So ( ) 1,p n0 uW∈Ω and () () 1,p 1,p n 00 nn WW uuR,n ΩΩ = ≤∀∈ . By virtue of the reflexivity of the space ( ) 1,p 0 W Ω , there exists ( ) 1,p 0 uW ∈ Ω such that weak n uu→ in ( ) 1,p 0 W Ω for some subsequence. We will prove that u is a generalized solution of the equation (1.1) in ( ) 1,p 0 W Ω , i.e. ( ) ( ) ( ) ( ) ( ) ( ) ( ) c ax,ux, ux dx fx,ux, ux dx 0, C ∞ ΩΩ ∇ ∇ϕ + ∇ ϕ = ∀ϕ∈ Ω ∫∫ In order to do that, we need the following lemma: 3.2.Lemma 3.2. Let m in , we have TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 9, SỐ 9 -2006 Trang 33 ( ) ( ) [ ] m nn n n n lim a x,u , u a x,u , u u u dx 0 →∞ Ω ∇ −∇∇−∇=⎡⎤ ⎣⎦ ∫ Proof: We only need to consider n>m+1. Let m φ be a functions in () c C ∞ Ω , with m 01 ≤ φ≤ in Ω and () m m m+1 1 if x x 0 if x \ ∈Ω ⎧ φ= ⎨ ∈Ω Ω ⎩ . Then there exists M such that, ( ) ( ) mm xM, xM, x φ ≤∇φ ≤∀∈Ω (3.10) Put () nmn w.uu=φ − restricted on n Ω . Because ( ) mn m2 n supp . u u , n m 1 + φ−⊂Ω⊂Ω∀>+ ⎡⎤ ⎣⎦ , we have nn suppw , n m 1⊂Ω ∀ > + . So ( ) 1,p n0n wW∈Ω. Since u n is the solution of the equation (3.2), we have ( ) ( ) nn nn n nnn ax,u, u wdx fx,u, u wdx 0 ΩΩ ∇ ∇+ ∇ = ∫∫ . Hence ( ) ( ) m1 m1 nn n nnn ax,u, u wdx fx,u, u wdx 0 ++ ΩΩ ∇∇ + ∇ = ∫∫ (3.11) We shall prove that ( ) ( ) m1 nnmn n lim f x, u , u u u dx 0 + →∞ Ω ∇ φ− = ∫ (3.12) by finding a number s such that ( ) nn f.,u, u∇ is bounded in ( ) s' m1 L + Ω and n uu→ in ( ) s m1 L + Ω . Since * [0,p 1)β∈ − and p<p * , we can find s satisfying * 1sp β +<< and p<s. Hence s s1 s' β< − = and pp p1p ss' γ< − < − = . Since { } n u is bounded in ( ) 1,p 0 W Ω , the Sobolev imbedding implies that there exists a subsequence still denoted by { } n u such that n uu→ in ( ) s L Ω , so n uu→ in ( ) s m1 L + Ω . From (1.5) and Lemma 2.1, one deduces that ( ) nn f.,u, u∇ is bounded in ( ) s' m1 L + Ω . Combining (3.10), we have (3.12). Hence ( ) m1 nnn n lim f x,u , u w dx 0 + →∞ Ω ∇ = ∫ (3.13) From (3.11), (3.13), one deduces that ( ) m1 nn n n lim a x, u , u w dx 0 + →∞ Ω ∇ ∇= ∫ or ( ) ( ) ( ) m1 nn mn n lim a x, u , u . u u dx 0 + →∞ Ω ∇ ∇φ − = ∫ Hence, ( ) ( ) ( ) m1 nnm n n m n lim ax,u,u .uu uu. dx0 + →∞ Ω ∇ φ∇ − + − ∇φ =⎡⎤ ⎣⎦ ∫ (3.14) Besides, since p < s, we get n uu→ in ( ) p m1 L + Ω (3.15) Applying Lemma 2.1, we have ( ) nn a.,u, u∇ is bounded in () N p' m1 L + ⎡ ⎤ Ω ⎣ ⎦ . Combining with (3.10), (3.15), we obtain ( ) ( ) m1 nnn m n lim a x, u , u u u . dx 0 + →∞ Ω ∇ −∇φ = ∫ (3.16) From (3.14), (3.16) we have ( ) ( ) m1 nnm n n lim a x,u , u . u u dx 0 + →∞ Ω ∇ φ∇ − = ∫ (3.17) On the other hand, (1.2), Lemma 2.1, (3.15) imply Science & Technology Development, Vol 9, No.9- 2006 Trang 34 ( ) ( ) n a.,u, u a.,u, u ∇ →∇ in () N p' m1 L + ⎡ ⎤ Ω ⎣ ⎦ Combining with (3.10) and the boundedness of un in ( ) 1,p 0m1 W + Ω , one deduces that ( ) ( ) ( ) m1 nmn n lim a x,u , u a x,u, u . u u dx 0 + →∞ Ω ∇ −∇φ∇−=⎡⎤ ⎣⎦ ∫ (3.18) and due to the weak convergence of un to u in ( ) 1,p 0m1 W + Ω also ( ) ( ) m1 mn n lim a x,u, u . u u dx 0 + →∞ Ω ∇ φ∇ − = ∫ (3.19) It follows from (3.18), (3.19) that ( ) ( ) m1 nmn n lim a x, u , u . u u dx 0 + →∞ Ω ∇ φ∇ − = ∫ (3.20) Hence (3.17) together with (3.20) yield ( ) ( ) ( ) m1 nn n m n n lim a x,u , u a x,u , u . u u dx 0 + →∞ Ω ∇ −∇φ∇−=⎡⎤ ⎣⎦ ∫ Since ( ) ( ) ( ) mnn n n . a x,u , u a x,u , u u u 0φ∇−∇∇−∇≥ ⎡⎤ ⎣⎦ , for all x in m1+ Ω , we get ( ) ( ) ( ) m nn n m n n lim ax,u, u ax,u, u . u udx 0 →∞ Ω ∇ −∇φ∇−=⎡⎤ ⎣⎦ ∫ The fact that ( ) mm x1,x φ =∀∈Ω then implies ( ) ( ) ( ) m nn n n n lim a x,u , u a x,u , u u u dx 0 →∞ Ω ∇ −∇∇−=⎡⎤ ⎣⎦ ∫ Fix () c C ∞ ϕ∈ Ω , there exists m in such that m supp ϕ ⊂Ω . Applying Lemma 2.5, Lemma 3.2, we have n uu∇→∇in ( ) p m L Ω for some subsequence. Since n uu→ in () p m L Ω also, together with Lemma 2.1, we obtain ( ) ( ) mm nn ax,u, u dx ax,u, u dx ΩΩ ∇ ∇ϕ → ∇ ∇ϕ ∫∫ ( ) ( ) mm nn f x,u , u dx f x,u, u dx ΩΩ ∇ ϕ→ ∇ϕ ∫∫ So ( ) ( ) ( ) ( ) mm a x,u, u dx f x,u, u dx a x,u, u dx f x,u, u dx 0 ΩΩΩ Ω ∇∇ϕ+ ∇ϕ = ∇∇ϕ+ ∇ϕ = ∫∫∫ ∫ Therefore, we get the main theorem: Theorem 3.1. Under the conditions (1.2)-(1.6), equation (1.1) has at least a generalized solution u in () 1,p 0 W Ω , that is, for any ( ) c C ∞ ϕ ∈Ω TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 9, SỐ 9 -2006 Trang 35 ( ) ( ) ax,u,u dx fx,u,u dx 0 ΩΩ ∇ ∇ϕ + ∇ ϕ = ∫∫ SỰ TỒN TẠI NGHIỆM CỦA PHƯƠNG TRÌNH ELLIPTIC QUASILINEAR VỚI ĐIỀU KIỆN KÌ DỊ Chung Nhân Phú, Trần Tấn Quốc Trường Đại học Khoa học tự nhiên, ĐHQG-HCM TÓM TẮT: Trong bài báo này, chúng tôi khảo sát sự tồn tại nghiệm suy rộng của một lớp phương trình elliptic kì dị: () () ( ) ( ) ( ) ( ) diva x,u x , u x f x,u x , u x 0−∇+∇= Sử dụng phương pháp xấp xỉ Galerkin trong [2,10] và hàm thử được Drabek, Kufner, Nicolosi nêu trong [5], chúng tôi mở rộng một số kết quả về phương trình elliptic trong [2,3,4,6,10]. REFERENCES [1]. Adams A., Sobolev spaces, Academic Press, (1975) [2]. Browder F. E., Existence theorem for nonlinear partial differential equations, Pro.Sym. Pure Math., Vol XVI, ed. by Chern S. S. and Smale S., AMS, Providence, p 1-60, (1970). [3]. 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Skrypnik I.V., Methods for Analysis of Nonlinear Elliptic Boundary Value Problems , AMS (1994) . KH&CN, TẬP 9, SỐ 9 -2006 Trang 27 EXISTENCE OF SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS WITH SINGULAR CONDITIONS Chung Nhan Phu, Tran Tan Quoc University of Natural Sciences, VNU-HCM (Manuscript. about elliptic equations in [2, 3, 4, 6, 10]. 1.INTRODUCTION The aim of this paper is to prove the existence of generalized solutions in ( ) 1, 0 p W Ω for the quasilinear elliptic equations: . Revised October 2 nd , 2006) ABSTRACT: In this paper, we study the existence of generalized solution for a class of singular elliptic equation: ( ) ( ) ( ) ( ) ( ) ( ) diva x,u x , u x f x,u